Solve x/x+2+1/2x=0

$$\frac{ x }{ x+2 } + \frac{ 1 }{ 2x } =0$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{15}i}{4}\approx -0.25+0.968245837i$$

$$x=\frac{-\sqrt{15}i-1}{4}\approx -0.25-0.968245837i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of x+2,2x.

$$2xx+x+2=0$$

Multiply x and x to get x^{2}.

$$2x^{2}+x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\times 2}}{2\times 2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8\times 2}}{2\times 2}$$

Multiply -8 times 2.

$$x=\frac{-1±\sqrt{1-16}}{2\times 2}$$

Add 1 to -16.

$$x=\frac{-1±\sqrt{-15}}{2\times 2}$$

Take the square root of -15.

$$x=\frac{-1±\sqrt{15}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-1±\sqrt{15}i}{4}$$

Now solve the equation x=\frac{-1±\sqrt{15}i}{4} when ± is plus. Add -1 to i\sqrt{15}.

$$x=\frac{-1+\sqrt{15}i}{4}$$

Now solve the equation x=\frac{-1±\sqrt{15}i}{4} when ± is minus. Subtract i\sqrt{15} from -1.

$$x=\frac{-\sqrt{15}i-1}{4}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{15}i}{4} x=\frac{-\sqrt{15}i-1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of x+2,2x.

$$2xx+x+2=0$$

Multiply x and x to get x^{2}.

$$2x^{2}+x+2=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}+x=-2$$

Divide both sides by 2.

$$\frac{2x^{2}+x}{2}=-\frac{2}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{1}{2}x=-\frac{2}{2}$$

Divide -2 by 2.

$$x^{2}+\frac{1}{2}x=-1$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=-1+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=-1+\frac{1}{16}$$

Add -1 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=-\frac{15}{16}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{4}\right)^{2}=-\frac{15}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{15}i}{4} x+\frac{1}{4}=-\frac{\sqrt{15}i}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{-1+\sqrt{15}i}{4} x=\frac{-\sqrt{15}i-1}{4}$$